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1

Mean-field optimal control problem of SDDES driven by fractional Brownian motion

Douissi Soukaina, Agram Nacira, Hilbert Astrid

Probability and Mathematical Statistics

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2020

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Vol. 40, Fasc. 1

139--158

EN

We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than 1/2. Stochastic optimal control problems driven by fractional Brownian motion cannot be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional white noise calculus combined with some special tools related to differentiation for functions of measures, we establish necessary and sufficient stochastic maximum principles. To illustrate our study, we consider two applications: we solve a problem of optimal consumption from a cash flow with delay and a linear-quadratic (LQ) problem with delay.

2

On the Distribution and q-Variation of the Solution to the Heat Equation with Fractional Laplacian

Khalil Z. Mahdi, Tudor C. A.

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 2

315--335

EN

We study the probability distribution of the solution to the linear stochastic heat equation with fractional Laplacian and white noise in time and white or correlated noise in space. As an application, we deduce the behavior of the q-variations of the solution in time and in space.

3

Occupation time problem for multifractional Brownian motion

Ouahra Mohamed Ait, Guerbaz Raby, Ouahhabi Hanae, Sghir Aissa

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

99--113

EN

In this paper, by using a Fourier analytic approach, we investigate sample path properties of the fractional derivatives of multifractional Brownian motion local times. We also show that those additive functionals satisfy a property of local asymptotic self-similarity. As a consequence, we derive some local limit theorems for the occupation time of multifractional Brownian motion in the space of continuous functions.

4

Basic characteristics of networks with self-similar traffic simulation

Aleksander Marek, Odarchenko Roman, Gnatyuk Sergiy, Kantor Tadeusz

Autobusy : technika, eksploatacja, systemy transportowe

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2019

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R. 20, nr 1-2

137--141

EN

This paper is devoted to simulations the networks with self-similar traffic. The self-similarity in the stochastic process is identified by calculation of the herst parameter value. Based on the results, received from the experimental research of network perfomance, we may conclude that the observed traffic in real-time mode is self-similar by its nature. Given results may be used for the further investigation of network traffic and work on the existing models of network traffic (particularly for new networks concepts like IoT, WSN, BYOD etc) from viewpoint of its cybersecurity. Furthermore, the adequacy of the description of real is achieved by complexifying the models, combining several models and integration of new parameters. Accordingly, for more complex models, there are higher computing abilities needed or longer time for the generation of traffic realization.

5

Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion

Borkowski D., Jańczak-Borkowska K.

Opuscula Mathematica

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2018

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Vol. 38, no. 3

307--326

EN

We study the existence and uniqueness of the backward stochastic variational inequalities driven by m-dimensional fractional Brownian motion with Hurst parameters Hk (k = 1,... m) greater than 1/2. The stochastic integral used throughout the paper is the divergence type integral.

6

Remarks on Pickands’ theorem

Michna Z.

Probability and Mathematical Statistics

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2017

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Vol. 37, Fasc. 2

373--393

EN

In this article we present the Pickands theorem and his double sum method. We follow Piterbarg’s proof of this theorem. Since his proof relies on general lemmas, we present a complete proof of Pickands’ theorem using the Borell inequality and Slepian lemma. The original Pickands’ proof is rather complicated and is mixed with upcrossing probabilities for stationary Gaussian processes. We give a lower bound for Pickands constant. Moreover, we review equivalent definitions, simulations and bounds of Pickands constant.

7

Cross-variation of Young integral with respect to long-memory fractional Brownian motions

Nourdin I., Zintout R.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 1

35--46

EN

We study the asymptotic behaviour of the cross-variation of two-dimensional processes having the form of a Young integral with respect to a fractional Brownian motion of index H > 1/2 . When H is smaller than or equal to 3/4 , we show asymptotic mixed normality. When H is stricly greater than 3/4 , we obtain a limit that is expressed in terms of the difference of two independent Rosenblatt processes.

8

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

Alpay D., Kipnis A.

Opuscula Mathematica

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2013

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Vol. 33, no. 3

395--417

EN

Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

9

Fractional white noise perturbations of parabolic Volterra equations

Sperlich S., Wilke M.

Journal of Applied Analysis

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2010

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Vol. 16, nr 1

31-48

EN

Aim of this work is to extend the results of Clément, Da Prato and Prüss [5] on the fractional white noise perturbation with Hurst parameter H ∈ (0,1). We will obtain similar results and it will turn out that the regularity of the solution u(t) increases with Hurst parameter H.

10

Small deviation of subordinated processes over compact sets

Linde W., Zipfel P.

Probability and Mathematical Statistics

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2008

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Vol. 28, Fasc. 2

281--304

EN

Let A =(A(t)t≥0 be a subordinator. Given a compact set K ⊂[(0;∞) we prove two-sided estimates for the covering numbers of the random set {A(t) : t ∈ K} which depend on the Laplace exponent Φ of A and on the covering numbers of K. This extends former results in the case K = [0; 1]. Using this we find the behavior of the small deviation probabilities for subordinated processes(WH(A(t))tЄK, whereWH is a fractional Brownian motion with Hurst index 0 < H < 1. The results are valid in the quenched as well as in the annealed case. In particular, those questions are investigated for Gamma processes. Here some surprising new phenomena appear. As application of the general results we find the behavior of log P(suptЄK |Zα(t)| < ε) as ε→ 0 for the α-stable Lévy motion Zα. For example, if K is a self-similar set with Hausdorff dimension D > 0, then this behavior is of order −ε−αD in complete accordance with the Gaussian case α = 2.

11

Occupation time fluctuations of Poisson and equilibrium finite variance branching systems

Miłoś P.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 2

181--203

EN

Functional limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in R dwith symmetric а-stable motion starting off from either a standard Poisson random field or from the equilibrium distribution for intermediate dimensions a < d < 2a. The limit processes are determined by sub-fractional and fractional Brownian motions, respectively.

12

Numerical solution of portfolio optimal control with constraints: the case of the object as weakly singular integral equation

Filatova D., Grzywaczewski M.

Journal of Applied Computer Science

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2007

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Vol. 15, nr 2

51-69

EN

We consider the model of optimal portfolio of Mertons’ market model. The noises involved in the dynamics of the wealth are fractional white noises. The stochastic optimal control problem is converted into a non-random optimization. An example of problem numerical solution illustrates proposed methodology.

13

Simulation of Pickands constants

Burnecki K., Michna Z.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 1

193--199

EN

Pickands constants appear in the asymptotic formulas for extremes of Gaussian processes. The explicit formula of Pickands constants does not exist. Moreover, in the literature there is no numerical approximation. In this paper we compute numerically Pickands constants by the use of change of measure technique. To this end we apply two different algorithms to simulate fractional Brownian motion. Finally, we compare the approximations with a theoretical hypothesis and a recently obtained lower bound on the constants. The results justify the hypothesis.

14

Asymptotics of the supremum of scaled Brownian motion

Dębicki K.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 1

199--212

EN

We consider the problem of estimating the tail of the distribution of the supremum of scaled Brownian motion B(ƒ(t)) processes with linear drift.Using the local time technique we obtain asymptotics and bounds of Pt≥t0(sup(B(ƒ(t))−t)> u), which are expressed in terms of the expected value of thelocal timeof B(ƒ(t))−tprocesses at levelu.As an application we obtain upper bounds for the tail of distribution of the supremum for some Gaussian processes with stationary increments.

15

On the supremum from gaussian processes over infinite horizon

Dębicki Krzysztof, Michna Zbigniew, Rolski Tomasz

Probability and Mathematical Statistics

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1998

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Vol. 18, Fasc. 1

83--100

EN

In the paper we study the asymptotic of the tail of distribution function P(A(X,c) > x) for x→∞, where A(X, c) is the supremum of X (t)—ct over [0, ∞). In particular, X(t) is the fractional Brownian motion, a nonlinearly scaled Brownian motion or some integrated stationary Gaussian processes. For the fractional Brownian motion we give a stronger result than a recent one of Duffield and O’Connell [5].